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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Saari's homographic conjecture of the three-body problem


Authors: Florin Diacu, Toshiaki Fujiwara, Ernesto Pérez-Chavela and Manuele Santoprete
Journal: Trans. Amer. Math. Soc. 360 (2008), 6447-6473
MSC (2000): Primary 70F10, 70H05
Published electronically: May 29, 2008
MathSciNet review: 2434294
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Abstract: Saari's homographic conjecture, which extends a classical statement proposed by Donald Saari in 1970, claims that solutions of the Newtonian $ n$-body problem with constant configurational measure are homographic. In other words, if the mutual distances satisfy a certain relationship, the configuration of the particle system may change size and position but not shape. We prove this conjecture for large sets of initial conditions in three-body problems given by homogeneous potentials, including the Newtonian one. Some of our results are true for $ n\ge 3$.


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Additional Information

Florin Diacu
Affiliation: Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada
Email: diacu@math.uvic.ca

Toshiaki Fujiwara
Affiliation: College of Liberal Arts and Sciences, Kitasato University, 1-15-1 Kitasato, Sagamihara, Kanagawa 228-8555, Japan
Email: fujiwara@clas.kitasato-u.ac.jp

Ernesto Pérez-Chavela
Affiliation: Departamento de Matemáticas, UAM–Iztapalapa, A.P. 55–534, 09340 Iztapalapa, Mexico, D.F., Mexico
Email: epc@xanum.uam.mx

Manuele Santoprete
Affiliation: Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada
Email: msantoprete@wlu.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04517-0
PII: S 0002-9947(08)04517-0
Keywords: Three-body problem, homographic solutions, central configurations
Received by editor(s): November 27, 2006
Published electronically: May 29, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.